Model Developments

“Uncertainty propagation is often based on strong assumptions regarding the probability distributions, which are mostly chosen for the sake of theoretical and numerical convenience rather than deduced from a probabilistic reasoning” [Guilleminot and Soize 2013a]. Hence, such models are not reliable. To circumvent this issue and increase the reliability of constructing a probabilistic model for the random coordinates, theory of random matrices and Maximum Entropy (MaxEnt) principal are employed.

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MaxEnt principal is a stochastic optimization procedure utilizing the Information Theory [Shannon 1948a and 1948b]. Using this principal beside some available information which is formed in a set of constraints one can construct the probability distributions of random coordinates.

Here we summarize what is done in Guilleminot and Soize (2013b) to model a 𝕄₆^ti(ℝ)-valued random variable:

6.2.2 Probabilistic Modeling of 𝕄_𝟔^𝒕𝒊(ℝ) Random Variable

Let [𝐂_{𝑐𝑙𝑎𝑦}^𝑡𝑖 ], matrix representation of stochastic consolidated clay elasticity tensor, be a 𝕄₆^𝑡𝑖(ℝ)-valued second-order random variable, and [𝑵] be the auxiliary 𝕄₆^𝑡𝑖 (ℝ)-valued random variable such that

[𝐂_{𝑐𝑙𝑎𝑦}^𝑡𝑖 ] = (𝔼{[𝐂_{𝑐𝑙𝑎𝑦}^𝑡𝑖 ] })^1/2[𝑵](𝔼{[𝐂_{𝑐𝑙𝑎𝑦}^𝑡𝑖 ] })^1/2 (6.5) And

𝔼{[𝑵]} = [𝐼], (6.6) 𝔼{log(det([𝑵]))} = 𝜈, |𝜈| < +∞ (6.7) Following [Guilleminot and Soize 2013b] there exists a matrix [𝑮] ([𝑮] is a random matrix which is unique and symmetric) such that

[𝑵] ∶= expm([𝑮]), (6.8) where expm is the matrix exponential, and

[𝑮] = ∑⁵_𝑖=1𝐺_𝑖[𝐸_ti^(𝑖)], (6.9) Now, we define 𝑮, {𝑮 ∶= (𝐺₁, 𝐺₂, 𝐺₃, 𝐺₄, 𝐺₅)}, a ℝ⁵ −valued random variable of coordinates on 𝕄₆^ti(ℝ). In the next part, the construction of marginal probability

46

distributions of random coordinate 𝑮 (or equivalently random matrix [𝑮]) is addressed.

Having the probability density functions (pdfs) of 𝑮, one can construct the pdf of coordinates of [𝐂_{𝑐𝑙𝑎𝑦}^𝑡𝑖 ] using Eq. (6.9) and (6.5).

6.2.3 Constructing a Probabilistic Model for Random Variable G

As mentioned earlier, the construction of probability distributions of random variable 𝑮 is completely equivalent to the construction of the one for random matrix [𝑮].

Let define {𝒈 → 𝑝_𝑮(𝒈)} as the family of marginal pdfs of 𝑮. Two steps are taken in deriving the stochastic model for random variable𝑮:

Step 1) Constructing the density p_G(g)

For the MaxEnt formulation, following constraints are considered based on information on [𝑮]:

𝔼 {∑⁵_𝑖=1𝐺_𝑖[𝐸_ti^(𝑖)]} = [𝐼], (6.10)

𝔼 {log (det (expm (∑⁵_𝑖=1𝐺_𝑖[𝐸_ti^(𝑖)])))} = 𝜈, |𝜈| < +∞ (6.11) By substituting (6.8)-(6.9) in (6.6)-(6.7) one can derive above equations. It can be inferred from Eq. (6.11) that both [𝑵] and [𝑵]⁻¹ are second-order random variables [Guilleminot and Soize 2013b]:

𝔼{‖[𝑵]‖_𝐹²} < +∞ 𝔼{‖[𝑵]⁻¹‖_𝐹²} < +∞ (6.12) Where ‖[𝐴]‖_𝐹 is Frobenius norm defined as: as ‖[𝐴]‖_𝐹 ≔≪ [𝐴], [𝐵] ≫ ^1/2.

For the self-readability, we recall here the MaxEnt principle. Let 𝐶_ad represents the set of all the integrable functions from 𝑺 ⊂ℝ⁵(𝑺 is the support of 𝑮) into ℝ⁺ such that satisfies above constraints and let 𝜀(𝑝) denotes the Shannon measure of entropy of pdf 𝑝:

47

𝜀(𝑝) = − ∫ 𝑝_{𝑺 𝐆}𝑙 𝑛(𝑝_𝐆) 𝑑𝒈 (6.13) The MaxEnt then reads:

𝑝_𝐆 = arg max 𝜀(𝑝) (6.14) 𝑝є 𝐶_ad

This method is utilized to obtain the most unbiased probabilistic model for 𝑝_𝐆. Following [Guilleminot and Soize 2013b], the general solution for the optimization problem (Eq. (6.14)) is obtained as:

𝑝_𝑮(𝒈) = 𝑐^𝑮 exp (−≪ [𝛬^sol], expm (∑⁵_𝑖=1𝑔_𝑖[𝐸_𝑡𝑖^(𝑖)]) ≫ – 𝜆^sol∑⁵_𝑖=1𝑔_𝑖 Tr([𝐸_𝑡𝑖^(𝑖)])) (6.15) where [𝛬^sol] and 𝜆^sol are the unknown Lagrange multipliers and must satisfy constraints (6.10) and (6.11). Following [Guilleminot and Soize 2013b], one can assume [𝛬^sol] =

∑⁵_𝑖=1𝜆_𝑖^sol[𝐸_𝑡𝑖^(𝑖)]. It is assumed that the optimization problem is well-posed and the it admits at most one solution. In the sequel, solution Lagrange multipliers will be considered as the vector-valued 𝝀^sol ∶= (𝜆₁^sol , . . . , 𝜆₅^sol , 𝜆^sol).

Step 2) Random generator and definition of 𝐺

Let define the potential function, ϕ, from ℝ⁵ into ℝ as [Guilleminot and Soize 2013b]:

ϕ(𝒖, 𝛌) = ≪ ∑⁵_𝑖=1𝜆^(𝑖)[𝐸_𝑡𝑖^(𝑖)], expm (∑⁵_𝑖=1𝑢_𝑖[𝐸_𝑡𝑖^(𝑖)]) ≫ – 𝜆 ∑⁵_𝑖=1𝑢_𝑖 Tr([𝐸_𝑡𝑖^(𝑖)]) (6.16) and let define 𝒁_𝝀 as the ℝ⁵-valued random variable with the pdf 𝑝_𝝀: ℝ⁵ → ℝ⁺ given by 𝑝_𝝀(𝒖) = 𝑐_𝝀 exp (−ϕ(𝒖)) (6.17) where 𝑐_𝝀 is a normalization constant. One can deduce that [Guilleminot and Soize 2013b]

𝑝_𝑮(𝒈) = 𝑝_𝝀^𝑠𝑜𝑙(𝒈) (6.18)

48 -valued normalized Wiener process. Following Guilleminot and Soize 2013b we obtain:

𝑟→+∞lim 𝑼(𝑟) = 𝒁_𝝀 (6.20) in probability distribution. From (6.18) and (6.20), one can define random variable 𝑮 as 𝑮 ∶= ℋ ({𝑾(𝑟), 𝑟>0}) , (6.21) Where ℋ is a non-linear operator.

Following Guilleminot and Soize 2013b, we make use of the Stormer-Verlet algorithm in order to discretize above ISDE to sample 𝑼(𝑟).

{ which are arbitrary deterministic vectors. The ℝ^𝑁-valued random variable 𝑳^𝑘 is defined as

49 (𝑳^𝑘)_𝑗 = −{^{𝜕ϕ(𝒖;𝛌)}

𝜕𝑢𝑗 }_𝒖=𝑼^𝑘 (6.23) Following Guilleminot and Soize 2013b, one can deduce that

𝑮 = lim

𝛥𝑟→0( lim

𝑘→+∞𝑼(𝑟_𝑘)) (6.24)

To generate the samples of 𝑼 (or as Eq. (6.24) suggests samples of 𝑮) one needs to find solutions of Lagrange multipliers. For the cubic and isotropic symmetry classes, explicit solutions of Lagrange multipliers can be easily constructed. This is not the case for transversely isotropic class of symmetry. Staber et al. 2015, propose a method to find the approximate solution of Lagrange multipliers which relies on sequential optimization problem. For the class of transversely isotropic symmetry, Lagrange multipliers are:

𝝀^sol = −0.8156(𝛿_[𝑵])^−2.01(−1, −1,0, −1, −1,1), (6.25) And subsequently

ϕ(𝒖) = 0.8156(𝛿_[𝑵])^−2.01X Tr(expm (∑⁵_𝑖=1𝑢_𝑖[𝐸_𝑡𝑖^(𝑖)]) − ∑⁵_𝑗=1𝑢_𝑗[𝐸_𝑡𝑖^(𝑗)]) (6.26) where 𝛿_[𝑵]∶= √𝔼{‖[𝑵] − 𝔼{[𝑵]}‖_𝐹²}/‖𝔼{[𝑵]}‖_𝐹² can be calculated using the available experimental data on elasticity matrix [𝐂_{𝑐𝑙𝑎𝑦}^𝑡𝑖 ] . By substituting (6.26) in (6.23) and using (6.22) one would be able to generate the samples of 𝑼and subsequently samples of 𝑮.

Now we aim to construct the pdf of coordinates {𝐶_𝑖^{𝑐𝑙𝑎𝑦}}

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This representation allows for simple algebraic calculation as shown in Walpole (1984).

Then, using Eq. (6.8), we obtain

[𝑵] ∶= expm([𝑮]) = {expm([𝐺₁₂₃]), exp(𝐺₄) , exp (𝐺₅)} (6.28) And finally using Eq. (6.5) one can generate multiple realizations of [𝑪] using a Monte Carlo simulation.

To find statistical dependency between coordinates{𝐶_𝑖}_𝑖=1⁵ , one can re-write Eq. (6.16) as:

𝑝_𝑮(𝒈) = 𝑝_{𝐺1, 𝐺2, 𝐺3}(𝑔₁, 𝑔₂, 𝑔₃) X 𝑝_𝐺4(𝑔₄) X 𝑝_𝐺5(𝑔₅) (6.29) Then, using the transformation (5.28) and (5.5) it is deduced that

𝑝_𝑪𝑐𝑙𝑎𝑦(𝒄^{𝑐𝑙𝑎𝑦}) = 𝑝_𝐶

1𝑐𝑙𝑎𝑦,𝐶₂^{𝑐𝑙𝑎𝑦},𝐶₃^{𝑐𝑙𝑎𝑦}(𝑐₁^{𝑐𝑙𝑎𝑦}, 𝑐₂^{𝑐𝑙𝑎𝑦}, 𝑐₃^{𝑐𝑙𝑎𝑦}) X 𝑝_𝐶

4𝑐𝑙𝑎𝑦(𝑐₄^{𝑐𝑙𝑎𝑦}) X 𝑝_𝐶

5𝑐𝑙𝑎𝑦(𝑐₅^{𝑐𝑙𝑎𝑦}) (6.30) which implies the following:

For the elasticity tensor of consolidated clay belonging to the transversely isotropic class of symmetry, the random coordinates {𝐶_𝑖^{𝑐𝑙𝑎𝑦}}

𝑖=1

In this section, we aim to construct the probabilistic model for the scalar sources of uncertainty (scalar random variables) in developed microporomechanics model. The

51

construction of such model is carried out by using the principle of Maximum Entropy (MaxEnt) which has been used in the previous section. Let assume that we aim to construct a probability model for uncertain real-valued parameter 𝑥. Then 𝑋 represents a real-valued random variable with a probability law defined by a pdf 𝑥 → 𝑝_𝑋(𝑥) on ℝ which we aim to construct. Following the MaxEnt principal

𝜀(𝑝) = − ∫ 𝑝_X𝑙 𝑛(𝑝_X) 𝑑𝑥

𝑺 (6.31) Thus, 𝑝_X(𝑥) is the pdf of random variable X which probabilistically models the uncertain variable 𝑥. Let assume that the available information on 𝑥 is (𝑖) the support of 𝑋, 𝑆_X= [𝑆₁, 𝑆₂]; (𝑖𝑖) the mean value of 𝑋, 𝜇_𝑋; and (𝑖𝑖𝑖) the dispersion of 𝑋, 𝛿_𝑋 = ^𝜎𝑋

𝜇𝑋 , where 𝜎_𝑋 represents the standard deviation of 𝑋. One can form following equations using available information on 𝑥 besides using the normalization condition of the 𝑝_X(𝑥):

∫ 𝑝_{𝑆 X}(𝑥)𝑑𝑥 The MaxEnt consists of maximizing the entropy in Eq. (6.31) subjected to the constraints (6.32)-(6.34). The ensuing pdf turns out to be

𝑝_𝑋^𝛌(𝑥) = 𝟙_𝑆(𝑥) exp (−λ₀− λ₁𝑥 − λ₂𝑥²) (6.35) where 𝟙_𝑆(𝑥) is the characteristic function of 𝑆, i.e. 𝟙_𝑆(𝑥) = 1 if 𝑥 ∈ 𝑆,0 otherwise; and vector 𝛌 = (λ₀, λ₁, λ₂) contains the Lagrange multipliers satisfying constraints in Eqs.

(44)-(46). 𝛌 is achieved by minimizing the Hamilton function defined as:

ℋ(𝛌) = λ₀+ λ₁𝜇_𝑋+ λ₂(1 + 𝛿_𝑋²) 𝜇_𝑋²+ ∫ exp (−λ_𝑆 0− λ₁𝑥 − λ₂𝑥²) 𝑑𝑥

𝑋 (6.36)

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6.3.1 Random Generator of 𝑿

In order to propagate the uncertainty of the uncertain parameter 𝑥, we should generate a sufficiently large number, 𝑁, of statistically independent realizations of 𝑋. To this end, we make use of pseudoinverse method [Devroye L. 1986] to generate realizations of random variable 𝑋 by utilizing the pdf calculated in the previous part. The N statistically independent realizations of random variable 𝑋, 𝑋(𝑎_𝑖)(1 = 1,2, … , 𝑁) can be obtained as following:

𝑋(𝑎_𝑖) = (𝐹_𝑋^λ)⁻¹{𝑈(𝑎_𝑖)} (6.37) where 𝐹_𝑋^λ(𝑥) = ∫ 𝑝_𝑆^𝑥 _𝑋^𝛌(𝜉) d𝜉

1 is the c.d.f. (cumulative density function) of 𝑋 and 𝑈(𝑎_𝑖) is a realization of a uniform random variable 𝑈 with values in 𝑆_𝑋.

We construct the pdf and generate independent realizations of scalar uncertain parameters in our model using described methodology.

6.4 Uncertainty Propagation

The algorithm of constructing probabilistic multiscale model of organic rich shale is schematically illustrated in the Fig. 6.1. The construction of probabilistic multiscale model requires substitution of the deterministic vector of input parameters with the random vector of input parameters. Since the input parameters are random, the model outputs will be random as well. Let 𝑾_{𝑙𝑒𝑣𝑒𝑙 𝐼} and 𝑾_{𝑙𝑒𝑣𝑒𝑙 𝐼𝐼} be the random vectors of input parameters at level I and level II. Then, one can show that the elasticity tensor of homogenized medium at level I and level II can be shown by:

at level I:⟦𝑪⟧_ℎ𝑜𝑚^{𝑙𝑒𝑣𝑒𝑙 𝐼} = 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛₁^{(𝑾𝑙𝑒𝑣𝑒𝑙 𝐼)} => 𝑓₁(⟦𝑪⟧_ℎ𝑜𝑚^{𝑙𝑒𝑣𝑒𝑙 𝐼}; 𝑾_{𝑙𝑒𝑣𝑒𝑙 𝐼}) = 0 (6.38a)

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In order to solve the above system of stochastic equations, the Monte Carlo simulation is employed. At the first step, for each statistically independent realization of 𝑾_{𝑙𝑒𝑣𝑒𝑙 𝐼}, statistically independent realization of ⟦𝑪⟧_ℎ𝑜𝑚^{𝑙𝑒𝑣𝑒𝑙 𝐼} is obtained by solving the Eq.

(6.38a). Then, statically independent realizations of 𝑾_{𝑙𝑒𝑣𝑒𝑙 𝐼𝐼} are used in order to obtain the statically independent realizations of ⟦𝑪⟧_ℎ𝑜𝑚^{𝑙𝑒𝑣𝑒𝑙 𝐼𝐼} by solving Eq. (6.38b).

Finally, statistics on ⟦𝑪⟧_ℎ𝑜𝑚^{𝑙𝑒𝑣𝑒𝑙 𝐼} and ⟦𝑪⟧_ℎ𝑜𝑚^{𝑙𝑒𝑣𝑒𝑙 𝐼𝐼} are obtained by calculating the statistics on the statically independent realizations of ⟦𝑪⟧_ℎ𝑜𝑚^{𝑙𝑒𝑣𝑒𝑙 𝐼} and ⟦𝑪⟧_ℎ𝑜𝑚^{𝑙𝑒𝑣𝑒𝑙 𝐼𝐼} obtained from solving the Eqs. (6.38a) and (6.38b).

𝑓𝑜𝑟 𝑖𝑡𝑟_𝑖 = 1 𝑡𝑜 𝑁 (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑖𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛𝑠)

Compute the statically independent realizations of random input vectors

Calculate the elasticity tensor at level I and level II

𝑒𝑛𝑑

Fig. 6.1 Constructing probabilistic multiscale model of organic-rich shale through Monte Carlo simulation 𝑖𝑡𝑟_𝑖 Indep. random realization

𝑊 _{𝑙𝑒𝑣𝑒𝑙 𝐼}(𝑖𝑡𝑟_𝑖) 𝑊 _{𝑙𝑒𝑣𝑒𝑙 𝐼𝐼}(𝑖𝑡𝑟_𝑖)

𝑊 _{𝑙𝑒𝑣𝑒𝑙 𝐼}(𝑖𝑡𝑟_𝑖)

𝐸𝑞. (6.38𝑎) ⟦𝐶⟧_ℎ𝑜𝑚^{𝑙𝑒𝑣𝑒𝑙 𝐼}

𝑊 _{𝑙𝑒𝑣𝑒𝑙 𝐼𝐼}(𝑖𝑡𝑟_𝑖)

𝐸𝑞. (6.38𝑏) ⟦𝐶⟧_ℎ𝑜𝑚^{𝑙𝑒𝑣𝑒𝑙 𝐼𝐼}

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6.5 Chapter Summary

This chapter deals with the construction of a probabilistic multiscale model which quantifies the uncertainties in the uncertain input parameters, and propagates them through multiple length scales to capture the effect of uncertainties on the model output. To this end, a framework of maximum entropy principal and random matrix theory are employed to construct a probabilistic model for the uncertain parameters based on available information. Uncertain parameters are classified as random matrix and random scalar variable, and it is explained in detail how to construct a probabilistic model for both categories, and how to generate samples from those models. Finally, the uncertainties are propagated through different levels using generated samples of uncertain parameters and a Monte Carlo simulation.

55 7 RESULTS

This chapter is dedicated to the implementation of probabilistic multiscale model developed in previous chapters. First, multiscale model is calibrated using available datasets. Then, values obtained from calibration procedure are validated. Next, an example is given to clarify the construction of probabilistic model at different length scales. Finally, a sensitivity analysis is performed to quantify the contribution of uncertain input parameters to the model output at different length scales.

7.1 Model Calibration 7.1.1 Optimization Problems

The goal of this section is to identify the values of input parameters to our model which cannot be obtained through experiments, or their values cover a wide range and thus it is not easy to determine them based on existing data. Volume fraction of organic and inorganic phases at multiple levels, their corresponding stiffness tensors, the radius of inclusions’ grain, and thickness and elasticity properties of ITZ are the input parameters to our model. It is turned out that stiffness tensor of consolidated clay, bulk modulus and Poisson’s ratio of kerogen, and thickness and elasticity properties of ITZ cannot be accurately determined from available data. Although some attempts to obtain the stiffness tensor of clay minerals at level 0 have been reported [among others see Alexandrov and Ryzhova 1961; Katahara 1996; Wang et al. 2001], the large range of estimated values for stiffness of clay minerals makes it impossible to choose a value based on these data.

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Besides, based on reported values from Monfared and Ulm (2016) and Ortega et al. (2007), stiffness values of consolidated clay at level I is considerably different from those of a single clay mineral.

Thus, a two-step model calibration is performed to determine the mean values of aforementioned input parameters to our model. In step 1, mean values of five independent components of elasticity tensor of consolidated clay at level 0, bulk modulus and Poisson’s ratio of kerogen at level I are determined. In step 2, mean values of thickness and elasticity properties of ITZ are determined.

Step 1: In order to find the stiffness tensor of consolidated clay at level I (or equivalently determine five independent components of stiffness tensor of consolidated clay), and bulk and Poisson’s ratio of kerogen a downscaling approach is set up. Here, the objective function is to find the values of aforementioned parameters such that Frobenius norm between measured and predicted indentation moduli at level I of samples in CDS 1 would be minimized. The indentation moduli at level I read:

M₃ = 2√^{𝐶11𝐶33−𝐶132}

Following equation represents the minimization problem that is set up to calibrate the multiscale model:

57

where 𝑛₁ and 𝑛₂ are the total number of measurements in samples 108, 150, 151 and Fay with measured values of M₁ and M₃, respectively. Also 𝒅₁ = <

𝐶₁₁^{𝑐𝑙𝑎𝑦}, 𝐶₁₂^{𝑐𝑙𝑎𝑦}, 𝐶₁₃^{𝑐𝑙𝑎𝑦}, 𝐶₃₃^{𝑐𝑙𝑎𝑦}, 𝐶₄₄^{𝑐𝑙𝑎𝑦} , 𝐾^{𝑘𝑒𝑟𝑜𝑔𝑒𝑛}, 𝜈^{𝑘𝑒𝑟𝑜𝑔𝑒𝑛}> represents the degrees of freedom associated with the minimization problem. 𝐾^{𝑘𝑒𝑟𝑜𝑔𝑒𝑛} represents bulk modulus of kerogen and 𝜈^{𝑘𝑒𝑟𝑜𝑔𝑒𝑛} denotes Poisson’s ration of kerogen. The minimization problem is subjected to a set of constraints to insure the positive definiteness of clay stiffness tensor at level I.

These constraints are:

𝐶₁₁+ 𝐶₁₂+ 𝐶₃₃+ 𝜉 > 0 (7.3)

𝐶₁₁+ 𝐶₁₂+ 𝐶₃₃− 𝜉 > 0

𝐶₁₁− 𝐶₁₂> 0 𝐶₄₄> 0 where

𝜉 = √𝐶₁₁²+ 𝐶₁₂²+ 8𝐶₁₃²+ 𝐶₃₃²+ 2𝐶₁₁𝐶₁₂− 2𝐶₁₁𝐶₃₃− 2𝐶₁₂𝐶₃₃ (7.4) In order to perform the optimization problem, a global search in MATLAB using fmincon interior-point optimization algorithm is employed. Obtained values from this optimization problem are used in Step 2 of model calibration in order to find the mean values of thickness of ITZ and its elastic properties.

Step 2: To obtain thickness of ITZ, 𝛥_𝑖𝑡𝑧, and its elastic properties (i.e. bulk and shear moduli) a downscaling of macroscopic elasticity of samples in calibration data set 2 (CDS2) is performed. In our model it is assumed that, as mentioned before, ITZ has an isotropic homogeneous behavior whose bulk and shear moduli are set equal to 𝐾^𝑖𝑡𝑧=

58

𝐶^𝑖𝑡𝑧𝐾^𝑒 and 𝐺^𝑖𝑡𝑧 = 𝐶^𝑖𝑡𝑧𝐺^𝑒 where 𝐶^𝑖𝑡𝑧 is a coefficient between 0 and 1 and 𝑒 denotes equivalent inclusion (self-consistent mixture of quartz and calcite).

Following objective function is set up to obtain the optimum values of parameters:

𝑚𝑖𝑛_𝑑2(∑ ‖[𝐶]_{ℎ𝑜𝑚,𝑝𝑟𝑒𝑑}^{𝐼𝐼,𝑢𝑛} − [𝐶]_{𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑}^𝐼𝐼 ‖

𝐶𝐷𝑆 2 𝐹) (7.5)

where 𝑑₂ =< 𝛥_𝑖𝑡𝑧, 𝐶^𝑖𝑡𝑧 > represents the degrees of freedom associated with the optimization problem, [𝐶]_{ℎ𝑜𝑚,𝑝𝑟𝑒𝑑}^{𝐼𝐼,𝑢𝑛} denotes predicted undrained stiffness matrix at level II, and [𝐶]_{𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑}^𝐼𝐼 is the measured undrained stiffness matrix at level II obtained through UPV measurement.

7.1.2 Input Parameters to Optimization Problems

Implementation of optimization in step 1 requires the volume fraction of consolidated clay, 𝜂^{𝑐𝑙𝑎𝑦}, kerogen, 𝜂^{𝑘𝑒𝑟𝑜𝑔𝑒𝑛}, porosity, 𝜑^I, and the values of indentation moduli, 𝑀₁ and 𝑀₃, for all the samples presented at CDS1. Table 7.1 contains the values of indentation moduli for samples in CDS1. Samples 108, 150,151, and Fay are used to calibrate the model at level I. Also, for the volume fraction of different phases at level I for samples in CDS1 see Table 3.2.

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Table 7.1 Calibration Data Set 1 (CDS1) - Indentation Moduli at level I (Abedi et al. 2016a)

For the second optimization problem, Step 2, volume fractions of calcite and quartz grains, 𝑓^{𝑐𝑎𝑙𝑐𝑖𝑡𝑒} and 𝑓^{𝑞𝑢𝑎𝑟𝑡𝑧}, porosity, 𝜙^II, consolidated clay, 𝑓^{𝑐𝑙𝑎𝑦}, and kerogen, 𝑓^{𝑘𝑒𝑟𝑜𝑔𝑒𝑛}, at level II and measurements on macroscopic undrained stiffness tensor/matrix of samples in the CDS2 are required. Also, mean values obtained from first optimization problem in Step 1, namely five independent elasticity components of consolidated clay, bulk modulus of kerogen, and Poisson’s ratio of kerogen are input parameters to the second optimization problem. Furthermore, inclusion grain radius is another input parameter to this optimization problem. This parameter is obtained through Scanning Electron Microscope (SEM) images of Haynesville shale, and the average inclusion grain radius is considered to be 2𝜇𝑚 [Monfared and Ulm 2016]. Table 7.2 represents five independent components of macroscopic undrained stiffness tensor/matrix of samples B1, B2, and B5 which form CDS2.

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Table 7.2 Calibration Data Set 2 (CDS2) - UPV measurements at level II (Monfared and Ulm 2016)

Table 7.3 contains the volume fractions of phases at level II for samples in Table 7.2.

Table 7.3 Volume fraction of constituent phases for samples presented in CDS2 at level II

All the other volume fractions required for homogenization at level I and level II can be obtained using Table 3.2, Table 7.3 and formulas presented in chapter 5.

7.1.3 Optimization Result

Following the steps mentioned in section 7.1.1 and using the data provided in section 7.1.2 following values are obtained for parameters mentioned in section 7.1.1:

Result from Step 1:

𝐶₁₁^{𝑐𝑙𝑎𝑦} = 95.6 (GPa), 𝐶₁₂^{𝑐𝑙𝑎𝑦} = 49.6 (GPa), 𝐶₁₃^{𝑐𝑙𝑎𝑦} = 26.5 (GPa) 𝐶₃₃^{𝑐𝑙𝑎𝑦} = 56.8 (GPa), 𝐶₄₄^{𝑐𝑙𝑎𝑦} = 10.0 (GPa)

Sample B1 B2 B5

𝐂_𝟏𝟏 58.7 54.1 51.4

𝐂_𝟏𝟐 20.5 19.9 18.3

𝐂_𝟏𝟑 15.4 11.3 12.6

𝐂_𝟑𝟑 33.8 33.1 30.3

𝐂_𝟒𝟒 14.9 15.7 13.6

sample 𝒇^{𝒄𝒍𝒂𝒚} 𝒇^{𝒌𝒆𝒓𝒐𝒈𝒆𝒏} 𝒇^{𝒒𝒖𝒂𝒓𝒕𝒛} 𝒇^{𝒄𝒂𝒍𝒄𝒊𝒕𝒆} 𝝓^𝐈𝐈

B1 0.269 0.05 0.278 0.337 0.066

B2 0.335 0.067 0.246 0.279 0.073

B5 0.384 0.067 0.295 0.182 0.072

61 𝐾^{𝑘𝑒𝑟𝑜𝑔𝑒𝑛} = 6.35 (GPa), 𝜈^{𝑘𝑒𝑟𝑜𝑔𝑒𝑛} = 0.28 Result from Step 2:

𝛥_𝑖𝑡𝑧 = 0.5 (𝜇𝑚), 𝐶^𝑖𝑡𝑧 = 0.5

7.2 Model Validation

In order to validate the result obtained from model calibration, several steps are taken. First, the result obtained for five independent components of elasticity tensor at level 0 are compared to some data available in the literature. Next, result obtained from step 1 of model calibration are employed to predict the indentation moduli of samples in VDS1, and then obtained indentation moduli are compared to their measured counterparts.

Finally, result obtained from step 2 of model calibration combined with those from step 1 of model calibration are employed to predict the undrained elasticity tensor/matrix of samples in VDS2, and then are compared to their measured counterparts. Furthermore, an example is presented to validate the probabilistic developments and quantify the role of uncertainty in input parameters of the model on the model output at multiple length scales.

7.2.1 Validation of Consolidated Clay Elasticity Tensor at Level 0

In this section five independent components of stiffness tensor of consolidated clay at level 0 obtained from model calibration are compared to the reported values in the literature. Table 7.4 contains the components of transversely isotropic clay obtained from a combination of experimental techniques. It is clear that except for the value obtained for 𝐶₁₁^{𝑐𝑙𝑎𝑦}from optimization, the values of other components obtained from optimization are compared well with the values reported in Table 7.4. Besides, one needs to take into

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consideration that values in Table 7.4 are for individual clay particles at level 0, and not for the consolidated clay at level 0. So, some deviations from stiffness values obtained experimentally are expected as observed by Monfared and Ulm (2016), and Ortega et al.

(2007).

Table 7.4 Five independent components of elasticity tensor/matrix of some of the clay particles

7.2.2 Validation of Optimization Result at Level I

Values obtained from optimization problem in Step 1 for bulk modulus and Poisson’s of kerogen agree well with the data reported in the literature. 𝐾^{𝑘𝑒𝑟𝑜𝑔𝑒𝑛} = 6.35 (GPa) is in the range of multiple values reported by Ahmadov et al (2009) and Zeszotarski et al. (2004) for bulk modulus of kerogen. Also, 𝜈^{𝑘𝑒𝑟𝑜𝑔𝑒𝑛} = 0.28 which is obtained from optimization in Step 1 agrees with the finding of Bousige et al. (2016) which suggests that kerogen’s Poisson’s ratio is nearly constant (𝜐 ≈ 0.25) irrespective to its density and state of maturity. To further validate the obtained values through optimization problems, Clay Type 𝑪_𝟏𝟏(GPa) 𝑪_𝟏𝟐(GPa) 𝑪_𝟏𝟑(GPa) 𝑪_𝟑𝟑(GPa) 𝑪_𝟒𝟒(GPa) Muscovite(Alexandrov and Ryzhova 1961) 178 42.4 14.5 54.9 12.2 Muscovite(Vaughan and Guggenheim

1986)

184.3 48.3 23.8 59.1 16

Kaolinite(Katahara 1996) 171.5 38.9 26.9 52.6 14.8

Muscovite(Seo et al. 1999) 250 60 35 80 35

Chlorite(Katahara 1996) 181.8 56.8 90.1 96.8 11.4

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measured indentation moduli (M₁ and M₃) are plotted against predicted values for samples in VDS1 which are presented in Table 7.5 as well as samples in CDS1.

Table 7.5 Validation Data Set 1 (VDS1) - Indentation Moduli at level I (Abedi et al. 2016a)

Table 7.6 contains the volume fractions of samples B2, B5 and B6 in VDS1 at level I. For samples 46 and 49 refer to Table 3.2.

Table 7.6 Volume fraction of constituent phases for samples presented in VDS1 at level I

Sample M1 M3

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Fig. 7.1 a) and b) depict this comparison. In these figures, horizontal axis denotes the predicted values and vertical axis denotes experimental measurement. In these graphs measured values denote the average of all the measurements for M₁ or M₃ for each sample.

The points with a blue color belong to the CDS1 and the points with a red color belong to VDS1. It is clear from Fig. 7.1 a) and b) that model predictions for M₁ and M₃ is reliable for samples in VDS1 and CDS1.

a) b)

Fig 7.1 a) Represents predicted indentation moduli against measured indentation moduli M1 for both CDS 1 and VDS 1. b) Depicts comparison between predicted and measured indentation moduli M3 for both CDS1 and VDS1.

7.2.3 Validation of Optimization Result at Level II

In order to validate the result obtained from optimization at level II, predicted components of undrained stiffness tensor are plotted against their counterparts which are obtained through UPV measurements for samples in VDS2, which are shown in Table 7.7, as well as for samples in CDS2.

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Table 7.7 Validation Data Set 2 (VDS2) - UPV measurements at level II (Monfared and Ulm 2016)

Volume fractions of samples in VDS2 are presented in Table 7.8.

Table 7.8 Volume fraction of constituent phases for samples presented in VDS2 at level II

Fig. 7.2 shows this comparison. In this figure, horizontal and vertical axes denote model prediction and measured values, respectively. Blue and red points represent different components of stiffness tensors for samples in CDS2 and VDS2, respectively.

Sample B3 B4 B6

𝐂_𝟏𝟏 49.9 64.6 58.52

𝐂_𝟏𝟐 13.4 20.3 18.5

𝐂_𝟏𝟑 10.4 21.4 11.6

𝐂_𝟑𝟑 41.9 58.7 35.1

𝐂_𝟒𝟒 15.3 20.7 14.6

sample 𝒇^{𝒄𝒍𝒂𝒚} 𝒇^{𝒌𝒆𝒓𝒐𝒈𝒆𝒏} 𝒇^{𝒒𝒖𝒂𝒓𝒕𝒛} 𝒇^{𝒄𝒂𝒍𝒄𝒊𝒕𝒆} 𝝓^𝐈𝐈

B3 0.103 0.034 0.154 0.663 0.046

B4 0.181 0.055 0.187 0.521 0.056

B6 0.364 0.069 0.276 0.215 0.076

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7.2.4 Validation of Probabilistic Model at Level I and Level II

In order to quantify the role of uncertainty in input parameters to the model outputs, first one needs to obtain the statistical representation of these parameters and generate realizations from them. To explain this step and as an example, the statistical representation of uncertain input parameters to the model are obtained for sample B6.

Stiffness tensor of consolidated clay, volume fractions of clay and kerogen at level I, bulk modulus and Poisson’s ratio of kerogen, volume fractions of calcite and quartz at level II, thickness of ITZ and coefficient of elasticity properties of ITZ (𝐶^𝑒) are considered as uncertain input parameters.

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Following chapter 6, one can easily generate realizations of components of stiffness tensor of consolidated clay. To this end, we substitute Eq. (6.26) into Eq. (6.23) to obtain ℝ⁵-valued random variable 𝑳^𝑘 (with components (𝑳^𝑘)_𝑗 {j = 1, 2, 3, 4, 5}) which is a function of 𝛿_[𝑵] and [𝐸_𝑡𝑖^(𝑖)] {𝑖 = 1, 2, 3, 4, 5}. To obtain [𝐸_𝑡𝑖^(𝑖)] we need to select 𝒏, the unit normal orthogonal to the plane of isotropy. It is considered that 𝒏=(0,0,1) and 𝛿_[𝑵]= 0.25. Selected value of 𝛿_[𝑵] represents the uncertainty associated with the elasticity tensor of particles in Table 7.4. Although reported values in Table 7.4 belong to some of the clay particles and not to the consolidated clay at level 0, it is assumed that the uncertainty in elasticity tensor of consolidated clay at level 0 is close to the uncertainty associated with

Following chapter 6, one can easily generate realizations of components of stiffness tensor of consolidated clay. To this end, we substitute Eq. (6.26) into Eq. (6.23) to obtain ℝ⁵-valued random variable 𝑳^𝑘 (with components (𝑳^𝑘)_𝑗 {j = 1, 2, 3, 4, 5}) which is a function of 𝛿_[𝑵] and [𝐸_𝑡𝑖^(𝑖)] {𝑖 = 1, 2, 3, 4, 5}. To obtain [𝐸_𝑡𝑖^(𝑖)] we need to select 𝒏, the unit normal orthogonal to the plane of isotropy. It is considered that 𝒏=(0,0,1) and 𝛿_[𝑵]= 0.25. Selected value of 𝛿_[𝑵] represents the uncertainty associated with the elasticity tensor of particles in Table 7.4. Although reported values in Table 7.4 belong to some of the clay particles and not to the consolidated clay at level 0, it is assumed that the uncertainty in elasticity tensor of consolidated clay at level 0 is close to the uncertainty associated with

In document A Probabilistic Approach for Multiscale Poroelastic Modeling of Mature Organic-Rich Shales (Page 56-103)